# A Langevin approach to lattice dynamics in a charge ordered polaronic system

## TL;DR

This paper introduces a Langevin approach to simulate finite temperature lattice dynamics in a charge-ordered polaronic system, accurately capturing thermodynamic and dynamical properties across different temperature regimes.

## Contribution

It presents a novel Langevin-based method for modeling lattice dynamics in the Holstein model, simplifying memory effects and matching Monte Carlo results while providing detailed dynamical insights.

## Key findings

- Accurately reproduces thermodynamic properties of charge order
- Identifies four temperature regimes with distinct lattice dynamics
- Provides detailed analysis of phonon and polaron behavior across temperatures

## Abstract

We use a Langevin approach to treat the finite temperature dynamics of displacement variables in the half-filled spinless Holstein model. Working in the adiabatic regime we exploit the smallness of the adiabatic parameter to simplify the memory effects and estimate displacement costs from an "instantaneous" electronic Hamiltonian. We use a phenomenological damping rate, and uncorrelated thermal noise. The low temperature state has checkerboard charge order (CO) and the Langevin scheme generates equilibrium thermodynamic properties that accurately match Monte Carlo results. It additionally yields the dynamical structure factor, $D({\bf q}, \omega)$, from the displacement field $x({\bf r}, t)$. We observe four regimes with increasing temperature, $T$, classified in relation to the charge ordering temperature, $T_c$, and the `polaron formation' temperature $T_P$, with $ T_c \ll T_P$. For $T \ll T_c$ the oscillations are harmonic, leading to dispersive phonons, with increasing $T$ bringing in anharmonic, momentum dependent, corrections. For $T \sim T_c$, thermal tunneling events of the $x({\bf r})$ field occur, with a propagating `domain' pattern at wavevector ${\bf q} \sim (\pi, \pi)$ and low energy weight in $D({\bf q}, \omega)$. When $T_c < T < T_P$, the disordered polaron regime, domain structures vanish, the dispersion narrows, and low energy weight is lost. For $T \gtrsim T_P$ we essentially have uncorrelated local oscillations. We propose simple models to analyse this rich dynamics.